Calculus Examples

$\sum_{n = 1}^{150} - 1 - (n - 1)$

Step 1

Simplify the summation.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- 1 - n - - 1$

Step 1.1.2

Multiply $- 1$ by $- 1$ .

$- 1 - n + 1$

Step 1.2

Combine the opposite terms in $- 1 - n + 1$ .

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Step 1.2.1

Add $- 1$ and $1$ .

$- n + 0$

Step 1.2.2

Add $- n$ and $0$ .

$- n$

Step 1.3

Rewrite the summation.

$\sum_{n = 1}^{150} - n$

Step 2

Factor $- 1$ out of the summation.

$(- 1) \sum_{n = 1}^{150} n$

Step 3

The formula for the summation of a polynomial with degree $1$ is:

$\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}$

Step 4

Substitute the values into the formula and make sure to multiply by the front term.

$(- 1) (\frac{150 (150 + 1)}{2})$

Step 5

Simplify.

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Step 5.1

Add $150$ and $1$ .

$- 1 \frac{150 \cdot 151}{2}$

Step 5.2

Multiply $150$ by $151$ .

$- 1 (\frac{22650}{2})$

Step 5.3

Divide $22650$ by $2$ .

$- 1 \cdot 11325$

Step 5.4

Multiply $- 1$ by $11325$ .

$- 11325$